the tau-leap method for simulating stochastic kinetic models

Approximate accelerated stochastic simulation ofchemically reacting systems

Colin Gillespie

October 16, 2013

Overview

Stochastic kinetic models

Issues with the Direct methodτ -leap method (Gillespie, 2001)

Leap conditionsMid-point estimation

Examples

Source code (R and LATEX of these slides):https://github.com/csgillespie/talks/

Stochastic kinetic models

A biochemical network is represented as a set of pseudo-biochemicalreactions:

u species & v reactions

Ri : p11X1 + p12X2 + · · ·+ p1uXuci−→ q11X1 + q12X2 + · · ·+ q1uXu

Stochastic rate constant ci .

Hazard/instantaneous rate: hi(Xt , ci) where Xt = (X1,t , . . . ,Xu,t) is thecurrent state of the system.

Under mass-action stochastic kinetics, the hazard function is proportional toa product of binomial coefficients, with

hi(Xt , ci) = ci

u∏j=1

(Xj,t

pij

).

Stochastic kinetic model

Describe the SKM by a Markov jump process (MJP)

The effect of reaction Rk is to change the value of each species Xi byqji − pji

The stoichiometry matrix S has elements sij = qji − pji

It can be shown that the time to the next reaction is

t ∼ Exp(h0(Xt , c)) where h0(Xt , c) =v∑

i=1

hi(Xi , ci)

and the reaction is of type i with probability hi(Xt , ci)/h0(Xt , c)

The process is easily simulated using the Direct method (Gillespiealgorithm)

Example: Lotka-Volterra system

R1: Prey reproduction

X1c1−−→ 2X1

R2: Prey death, predator reproduction

X1 + X2c2−−→ 2X2

R3: Predator death

X2c3−−→ ∅

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Stochastic realisations

Prey

Predator

Time

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X(0) = (100, 100)c = (0.5, 0.0025, 0.3)

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(x)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(x)

4 Reaction time: τ = −ln[U(0, 1)]/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to it’s hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically there are a large number of iterates

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(x)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(x)

4 Reaction time: τ = −ln[U(0, 1)]/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to it’s hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically there are a large number of iterates

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(x)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(x)

4 Reaction time: τ = −ln[U(0, 1)]/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to it’s hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically there are a large number of iterates

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(x)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(x)

4 Reaction time: τ = −ln[U(0, 1)]/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to it’s hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically there are a large number of iterates

Approximations

Relax some assumptions (e.g. discreteness and stochasticity) in order tomake simulation faster and more scalable

Diffusion approximation / chemical Langevin equation (CLE)

Linear noise approximation (LNA) / moment closure (2MA)

ODE

Hybrid discrete-continuous models

Poisson leap

If all reactions are zeroth-order, then the model is a homogeneousPoisson process

Hence the number of reactions in (t0, t1) follows a Poisson distributionFor a more general model, if we consider a small time interval,(t, t + ∆t), then:

the hazard rates should be approximately constantthe number of reactions (of a given type) can be sampled from a Poissondistribution

A balance between speed and accuracy

Poisson leap method

1 Set t = 0. Initialise the rate constants and the initial molecule numbers x2 Calculate hi(x , ci), for i = 1, . . . , v , and simulate the v -dimensional

reaction vector r , with i th entry a Po(hi(x , ci)∆t) random quantity3 Update the state according4 Update t := t + ∆t5 Output t and x . If t < Tmax return to step 2

How do you chose ∆t?

Poisson leap method

1 Set t = 0. Initialise the rate constants and the initial molecule numbers x2 Calculate hi(x , ci), for i = 1, . . . , v , and simulate the v -dimensional

reaction vector r , with i th entry a Po(hi(x , ci)∆t) random quantity3 Update the state according4 Update t := t + ∆t5 Output t and x . If t < Tmax return to step 2

How do you chose ∆t?

Example: Lotka-Volterra

Suppose we are interested inparameter inference

A possible prior could beindependent Uniform priorsover U(−8, 8) for each log(ci)

Three samples from this prioryield very different realisations

Probability of extinction bytime t = 30, is around 0.86

Each simulation would requirea very different ∆t

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Gillespie simulations

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Basic τ -leap method

Suppose a temporal leap τ will result in a state change λ

Choose a value of τ that satisfies the leap condition. For each reaction,Rj , we want

|hj(x + λ)− hj(x)|

to be small

Sample kj ∼ P(hj(x)τ)

Compute λ

Set t := t + τ and x := x + λ

Choosing τ

If the reactions don’t depend on x, the leap condition will be satisfiedexactly for any τ (and so exact)

If population numbers are large, then it would take a large number ofreactions to “noticeably” change the hazard functions

If satisfying the leap condition requires a very small value τ << 1/h0(x),then we may as well use the exact SSA

A procedure for selecting τ (Gillespie 2001)

The expected net change in (t, t + τ) will be:

λ̄ =v∑

j=1

[hj(x)τ ]sj = τξ(x)

So we require that the expected changes in the propensity functions intime τ , are bounded by some fraction of all propensity functions, i.e.

|hj(x + λ)− hj(x)| < εh0(x) for j = 1, . . . , v .

Estimate the difference using a Taylor expansion:

hj(x + λ)− hj(x) 'u∑

i=1

τξi(x)∂

∂xihj(x)

Defining

bji(x) ≡ ∂hj(x)

∂xi(j = 1, . . . , v ; i = 1, . . . , u)

A procedure for selecting τ

The requirement becomes

τ

∣∣∣∣∣u∑

i=1

ξi(x)bji(x)

∣∣∣∣∣ ≤ εh0(x) (j = 1, . . . , v)

The largest value of τ consistent with this condition (and hence optimal) is

τ = minj∈[1,v]

{εh0(x)∣∣∑u

i=1 ξi(x)bji(x)∣∣}

Typical values of ε are around 0.05

Example: Lotka-Volterra

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h1

h2

h3

Example: Lotka-Volterra

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Example: Lotka-Volterra

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Example: Lotka-Volterra

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Mean τ

τ

Freq

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0.0 0.1 0.2 0.3 0.4 0.5

The estimated-midpoint technique

The leap condition requires that the hazards functions do not“appreciably” change in the course of a leapBut we want to take large leaps, so we will inevitably get computationalerrorsThis is similar to solving the ODE

dX(t)dt

= f (X)

using an Euler schemeA standard technique is to use a second-order Runge-Kutta or modifiedEuler method

X(t + ∆t) = X(t) + f [X(t) + 0.5f (X(t))∆t]∆t

i.e. we use an Euler method to estimate the midpoint during [t, t + ∆t],then calculate the increment in X by evaluating the slope function f at thatestimated midpoint

Example: Death model

The death model contains a single reaction

X µ−−→ ∅

and has hazard function h1(x , µ) = µx and state change vector s = −1.

The solution to the CME is:

Pr(X = x ; t) =

(x0

x

)e−µτ(x0−x)(1− e−µτ )x

Death model

τ = 0.1 τ = 0.4 τ = 0.9

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Pr(k

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Method Exact

Pr(X = x ; τ) =

(x0

x

)e−µτ(x0−x)(1− e−µτ )x

Death model: p-leap

τ = 0.1 τ = 0.4 τ = 0.9

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Pr(k

)

Method Exact Poisson−leap

If we perform a single leap of length τ , the number of executed reactions are

Pp(k ;µx0, τ) =eµx0τ (µx0τ)k

k!for k = 0, . . .

Death model: τ -leap + mid-point

τ = 0.1 τ = 0.4 τ = 0.9

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Pr(k

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Method Exact Poisson−leap Mid−point

We estimate the mid-point to be:

x ′ ≡ x0 − b0.5τµx0c

so the number of reactions executed is

Pp(k ;µx0, τ) =eµx ′τ (µx ′τ)k

k!for k = 0, . . .

Example: Lotka-Volterra

Prey Predator

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p−leap τ−leap τ−leap + mid p−leap τ−leap τ−leap + mid

Simulator

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Example: Lotka-Volterra

Prey Predator

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p−leap τ−leap τ−leap + mid p−leap τ−leap τ−leap + mid

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For these parameter values and this model, we have the approximaterelationship (obtained via simulation)

ε ' 0.556×∆t

Example: Lotka-Volterra

Prey Predator

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Simulator

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For these parameter values and this model, we have the approximaterelationship (obtained via simulation)

ε ' 0.556×∆t

Summary

Similar issues arise when solving SDEs with the Euler-Maruyamascheme

In fact, if we substitute Poisson with Gaussian random numbers in thep-leap scheme, we get the Euler-Maruyama algorithm

Choosing a fixed ∆t for a wide range of parameter combinations doesn’tmake sense

Is it possible to these ideas when constructing bridges for SDEs?

References

Gillespie DT. Exact stochastic simulation of coupled chemical reactions. TheJournal of Physical Chemistry, 1977, 81: 2340 – 2361. Direct method

Gillespie, DT. Approximate accelerated stochastic simulation of chemicallyreacting systems. The Journal of Chemical Physics, 2001, 115: 1716. τ -leapmethod

Sandmann, W. Streamlined formulation of adaptive explicit-implicit tau-leapingwith automatic tau selection. Simulation Conference (WSC), Proceedings of the2009 Winter. IEEE, 2009. Nice overview of the various τ -leap flavours.

Golightly, A & Gillespie, CS. Simulation of Stochastic Kinetic Models. In SilicoSystems Biology. Humana Press, 2013, 169 – 187. Overview of various SSA.

https://github.com/csgillespie/In-silico-Systems-Biology